Giant Armadillo Optimization: A New Bio-Inspired Metaheuristic Algorithm for Solving Optimization Problems

In this paper, a new bio-inspired metaheuristic algorithm called Giant Armadillo Optimization (GAO) is introduced, which imitates the natural behavior of giant armadillo in the wild. The fundamental inspiration in the design of GAO is derived from the hunting strategy of giant armadillos in moving towards prey positions and digging termite mounds. The theory of GAO is expressed and mathematically modeled in two phases: (i) exploration based on simulating the movement of giant armadillos towards termite mounds, and (ii) exploitation based on simulating giant armadillos’ digging skills in order to prey on and rip open termite mounds. The performance of GAO in handling optimization tasks is evaluated in order to solve the CEC 2017 test suite for problem dimensions equal to 10, 30, 50, and 100. The optimization results show that GAO is able to achieve effective solutions for optimization problems by benefiting from its high abilities in exploration, exploitation, and balancing them during the search process. The quality of the results obtained from GAO is compared with the performance of twelve well-known metaheuristic algorithms. The simulation results show that GAO presents superior performance compared to competitor algorithms by providing better results for most of the benchmark functions. The statistical analysis of the Wilcoxon rank sum test confirms that GAO has a significant statistical superiority over competitor algorithms. The implementation of GAO on the CEC 2011 test suite and four engineering design problems show that the proposed approach has effective performance in dealing with real-world applications.


Introduction
There are many problems in mathematics, science, and real-world applications that have more than one feasible solution.These types of problems are known as optimization problems, and the process of obtaining the best feasible solution among all these existing solutions is called optimization [1].Each optimization problem is mathematically modeled using three main parts: decision variables, problem constraints, and an objective function.The goal in optimization is to allocate appropriate values for decision variables so that the objective function is optimized by respecting the constraints of the problem [2].There are numerous optimization problems in science, mathematics, engineering, technology, industry, and real-world applications that need to be solved using optimization techniques.Problem-solving techniques for solving optimization problems are classified into two classes: deterministic and stochastic approaches [3].Deterministic approaches in two categories, gradient-based and non-gradient-based, are effective in solving linear, convex, continuous, differentiable, and low-dimensional problems [4].However, as optimization problems become more complex, especially as the problem dimensions increase, deterministic approaches stop getting stuck in local optima [5].This is despite the fact that many practical optimization problems are non-linear, non-convex, non-differentiable, non-continuous, and high-dimensional.The disadvantages of deterministic approaches in order to solve practical optimization problems in science have led to researchers' efforts in designing stochastic approaches [6].
Metaheuristic algorithms are among the most efficient and well-known stochastic approaches that have been used to deal with numerous optimization problems.These algorithms are able to provide suitable solutions for optimization problems based on random search in the problem-solving space and benefit from random operators and trialand-error processes.The optimization mechanism in metaheuristic algorithms starts with the random generation of a certain number of candidate solutions under the name of algorithm population.Then, these candidate solutions are improved during successive iterations and based on the population update steps of the algorithm.After the full implementation of the algorithm, the best candidate solution obtained is presented as a solution to the problem [7].The nature of stochastic search results in no guarantee of definitively achieving the global optimum using metaheuristic algorithms.However, due to being close to the global optimum, the solutions obtained from metaheuristic algorithms are acceptable as pseudo-optimal [8].The desire of researchers to achieve more effective solutions closer to the global optimum for optimization problems has led to the design of numerous metaheuristic algorithms [9].These metaheuristic algorithms have been used to tackle optimization problems in various sciences, such as static optimization problems [10], green product design [11], feature selection [12], design for disassembly [13], image segmentation [14], and wireless sensor network applications [15].Metaheuristic algorithms will be able to achieve effective solutions for optimization problems when they search the problem-solving space well at both global and local levels.Global search expresses the exploration power of the algorithm in the extensive search in the problem-solving space with the aim of discovering the main optimal area and preventing the algorithm from getting stuck in local optima.Local search represents the exploitation power of the algorithm in the exact search near the promising areas of the problem-solving space and the discovered solutions.In addition to exploration and exploitation abilities, what leads to the success of a metaheuristic algorithm in providing a suitable solution for an optimization problem is their balancing during the search process in the problem-solving space [16].
The main research question is: according to the many metaheuristic algorithms designed so far, is there still a need to introduce newer metaheuristic algorithms in science or not?In response to this question, the No Free Lunch (NFL) [17] theorem explains that the successful performance of a metaheuristic algorithm in solving a set of optimization problems is no guarantee for the similar performance of that algorithm in solving other optimization problems.In fact, an algorithm may even converge to the global optimum in solving an optimization problem but fail in solving another problem by getting stuck in the local optimum.Therefore, there is no assumption about the failure or success of implementing a metaheuristic algorithm on an optimization problem.The NFL theorem explains that in no way can it be claimed that a unique metaheuristic algorithm is the best optimizer for all optimization problems.The NFL theorem, by keeping active the studies of metaheuristic algorithms, motivates researchers to be able to achieve more effective solutions for optimization problems by designing newer algorithms.
The innovation and novelty of this paper is the introduction of a new metaheuristic algorithm called Giant Armadillo Optimization (GAO) to solve optimization problems in various sciences.The main contributions of this study are as follows: • GAO is designed based on simulating the natural behavior of giant armadillos in the wild.• The fundamental inspiration for GAO is taken from the strategy of giant armadillos when attacking termite mounds.• The GAO theory has been described and mathematically modeled in two phases: (i) exploration based on simulating the movement of giant armadillos towards termite mounds, and (ii) exploitation based on simulating giant armadillos' digging skills in order to prey on and rip open termite mounds.• The performance of GAO is evaluated on the CEC 2017 test suite for problem dimen- sions of 10, 30, 50, and 100.• The performance of GAO in handling real-world applications is evaluated in handling twenty-two constrained optimization problems from the CEC 2011 test suite and four engineering design problems.• The results obtained from GAO are compared with the performance of twelve wellknown metaheuristic algorithms.
The proposed GAO approach has several advantages for global optimization problems.The first advantage of GAO is that there is no control parameter in the design of this algorithm, and therefore there is no need to control the parameters in any way.The second advantage of GAO is its high effectiveness in dealing with a variety of optimization problems in various sciences as well as complex, high-dimensional problems.The third advantage of the proposed GAO method is that it shows its great ability to balance exploration and exploitation in the search process, which allows it high-speed convergence to provide suitable values for decision variables in optimization tasks, especially in complex problems.The fourth advantage of the proposed GAO is its powerful performance in handling real-world optimization applications.
The structure of this paper is as follows: A literature review is presented in Section 2.Then, the proposed Giant Armadillo Optimization (GAO) is introduced and modeled in Section 3. Simulation studies and results are presented in Section 4. The effectiveness of GAO in solving real-world applications is investigated in Section 5. Conclusions and suggestions for future research are provided in Section 6.

Literature Review
Metaheuristic algorithms have been developed with inspiration from various natural phenomena, the behaviors of living organisms in the wild, genetic, biological, and physics sciences, game rules, human interactions, and other evolutionary phenomena.Metaheuristic algorithms are classified into five groups based on the main idea in design: swarm-based, evolutionary-based, physics-based, human-based, and game-based approaches.
Swarm-based metaheuristic algorithms are inspired by the lifestyles of animals, birds, insects, aquatics, reptiles, and other living creatures in the wild.The most well-known algorithms in this group are: Particle Swarm Optimization (PSO) [18], Ant Colony Optimization (ACO) [19], Artificial Bee Colony (ABC) [20], and Firefly Algorithm (FA) [21].PSO is inspired by the group movement of flocks of birds and fish towards food sources.ACO is inspired by the ability of ants to discover the optimal communication path between the colony and the food source.ABC is inspired by the activities of colony bees searching for food sources.FA is inspired by optical communication between fireflies.The Grey Wolf Optimizer (GWO) is a swarm-based metaheuristic algorithm that is inspired by the hierarchical leadership structure and social behavior of gray wolves during hunting [22].Green Anaconda Optimization (GAO) is inspired by the ability of male green anacondas to detect the position of females during the mating season and the hunting strategy of green anacondas [23].Among the natural behaviors of living organisms in the wild, foraging, hunting, digging, migration, and chasing are much more prominent and have been employed in the design of algorithms such as: Honey Badger Algorithm (HBA) [24], African Vultures Optimization Algorithm (AVOA), Whale Optimization Algorithm (WOA) [25], Orca Predation Algorithm (OPA) [26], Reptile Search Algorithm (RSA) [27], Kookaburra Optimization Algorithm (KOA) [28], Mantis Search Algorithm (MSA) [29], Liver Cancer Algorithm (LCA) [30], Marine Predator Algorithm (MPA) [31], Tunicate Swarm Algorithm (TSA) [32], White Shark Optimizer (WSO) [33], and Golden Jackal Optimization (GJO) [34].
Evolutionary-based metaheuristic algorithms are designed with inspiration from genetic and biological sciences, concepts of natural selection, survival of the fittest, Darwin's theory of evolution, and evolutionary operators.Genetic Algorithm (GA) [35] and Differential Evolution (DE) [36] are the most famous algorithms of this group, which are developed inspired by the reproduction process, genetic and biological concepts, and evolutionaryrandom operators of crossover, selection, and mutation.Artificial Immune Systems (AISs) are inspired by the mechanisms of the human body's immune system against microbes and diseases [37].Some other evolutionary-based metaheuristic algorithms are: Genetic programming (GP) [38], Cultural Algorithm (CA) [39], and Evolution Strategy (ES) [40].
Physics-based metaheuristic algorithms are designed with inspiration from the phenomena, forces, transformations, laws, and concepts of physics.Simulated Annealing (SA) is one of the most widely used algorithms of this group, which is inspired by the annealing process of metals, in which metals are first melted under heat, then slowly cooled with the aim of achieving an ideal crystal.Physical forces and Newton's laws of motion have been the source of design in algorithms such as the Momentum Search Algorithm (MSA) [41] based on momentum force, the Gravitational Search Algorithm (GSA) based on gravitational attraction force [42], and the Spring Search Algorithm (SSA) [43] based on the elastic force of the spring and Hooke's law.Cosmological concepts have been the origin of design in algorithms such as the Multi-Verse Optimizer (MVO) [44] and the Black Hole Algorithm (BHA) [45].Some other physics-based metaheuristic algorithms are: Archimedes Optimization Algorithm (AOA) [46], Water Cycle Algorithm (WCA) [47], Artificial Chemical Process (ACP) [48], Chemotherapy Science Algorithm (CSA) [49], Nuclear Reaction Optimization (NRO) [50], Henry Gas Optimization (HGO) [51], Electro-Magnetism Optimization (EMO) [52], Lichtenberg Algorithm (LA) [53], Thermal Exchange Optimization (TEO) [54], and Equilibrium Optimizer (EO) [55].
Human-based metaheuristic algorithms are designed with inspiration from thoughts, choices, decisions, communication, interactions, and other human activities in individual and social life.Teaching-Learning-Based Optimization (TLBO) is one of the most famous human-based metaheuristic algorithms, which is introduced with the inspiration of educational communication in the classroom environment and the exchange of knowledge between teachers and students and students with each other [56].The Mother Optimization Algorithm (MOA) is proposed based on the modeling of Eshrat's care of her children [57].Doctor and Patient Optimization (DPO) is introduced based on modeling the process of treating patients by doctors [58].Sewing Training-Based Optimization (STBO) is proposed with the inspiration of teaching sewing skills by the instructor to students in sewing schools [59].Ali Baba and the Forty Thieves (AFT) is presented based on modeling the strategies of forty thieves in the search for Ali Baba [60].Some other human-based metaheuristic algorithms are: Election-Based Optimization Algorithm (EBOA) [61], Coronavirus Herd Immunity Optimizer (CHIO) [62], Group Teaching Optimization Algorithm (GTOA) [63], Ebola Optimization Search Algorithm (ESOA) [64], Driving Training-Based Optimization (DTBO) [5], Gaining Sharing Knowledge-Based Algorithm (GSK) [65], and War Strategy Optimization (WSO) [66].
Game-based metaheuristic algorithms are inspired by the rules governing individual and team games and the strategies of players, coaches, referees, and other influential people in these games.Darts Game Optimizer (DGO) is one of the most well-known game-based metaheuristic algorithms, whose design is inspired by the strategy and skill of players in throwing darts and collecting points [67].Hide Object Game Optimizer (HOGO) is proposed based on the simulation of players' strategies for finding the hidden object on the playing field [68].The Orientation Search Algorithm (OSA) is designed based on modeling the players' position changes on the playing field based on the referee's commands [69].Some other game-based metaheuristic algorithms are: Ring toss game-based optimization (RTGBO) [70], Football Game Based Optimization (FGBO) [71], Archery Algorithm (AA) [6], Golf Optimization Algorithm (GOA) [72], and Volleyball Premier League (VPL) [73].
Based on the best knowledge obtained from the literature review, no metaheuristic algorithm inspired by the natural behavior of giant armadillos in nature has been designed so far.This is while the strategy of giant armadillos in attacking termite mounds and digging them is an intelligent process that has a special potential for designing a new optimizer.In order to address this research gap, a new bio-inspired metaheuristic algorithm is introduced in this paper based on the mathematical modeling of the strategy of giant armadillos in attacking and hunting in termite mounds, which is discussed in the next section.

Giant Armadillo Optimization
In this section, the source of inspiration in the design of the proposed Giant Armadillo Optimization (GAO) approach is stated, and then it is mathematically modeled in order to use it in optimization applications.

Inspiration for GAO
The giant armadillo (Priodontes maximus) is the largest living species of armadillo in danger of extinction and lives in South America, ranging as far south as northern Argentina [83].Termites and ants are the main diet of giant armadillos.However, this animal also feeds on plants, larvae, worms, and larger creatures, such as snakes and spiders.In order to feed on termites, giant armadilloes attack termite mounds and then use their digging power to prey on and rip open termite mounds.
The giant armadillo has 3 or 4 hinged bands protecting the neck and another 11 to 13 hinged bands that protect the body [84].Its body is dark brown with a lighter yellowish band along the sides, and its head is pale and yellowish-white.It also has very long front paws, up to 22 cm long.The tail is covered in small, rounded scales.The giant armadillo is almost entirely hairless.Giant armadillos weigh approximately 18.7-32.5kg, although specimens weighing 54 kg and 80 kg have also been observed.Their length without including the tail is between 75 and 100 cm, and the length of their tail is about 50 cm [85].An image of the giant armadillo is shown in Figure 1.
Among the natural behaviors of the giant armadillo, the strategy of this animal when it attacks termite mounds and then digs them with the aim of hunting and feeding on termites is much more prominent.Mathematical modeling of these two natural behaviors of giant armadillos during hunting, namely (i) attacking termite mounds and (ii) digging termite mounds in order to feed on them, has been employed in the design of the proposed GAO approach, which is discussed below.
Among the natural behaviors of giant armadillos, the hunting strategy of this animal is much more prominent.The giant armadillo hunting process has two stages: (i) moving towards termite mounds and (ii) digging in termite mounds in order to feed on termites.Mathematical modeling of these natural behaviors of the giant armadillo during hunting is employed in the design of the proposed GAO approach, which is discussed below.Among the natural behaviors of the giant armadillo, the strategy of this animal when it attacks termite mounds and then digs them with the aim of hunting and feeding on termites is much more prominent.Mathematical modeling of these two natural behaviors of giant armadillos during hunting, namely (i) attacking termite mounds and (ii) digging termite mounds in order to feed on them, has been employed in the design of the proposed GAO approach, which is discussed below.
Among the natural behaviors of giant armadillos, the hunting strategy of this animal is much more prominent.The giant armadillo hunting process has two stages: (i) moving towards termite mounds and (ii) digging in termite mounds in order to feed on termites.Mathematical modeling of these natural behaviors of the giant armadillo during hunting is employed in the design of the proposed GAO approach, which is discussed below.

Solution Process of the GAO
The proposed GAO approach is a biomimetics metaheuristic algorithm that mimics the natural behavior of the giant armadillo in the wild.Among the natural behaviors of the giant armadillo, the strategy of this animal in attacking termite mounds and then digging in them for feeding is employed in the GAO design.In this modeling, the wild life of the giant armadillo corresponds to the problem-solving space, and the position of each giant armadillo in the wild corresponds to the position of each GAO member in the problem-solving space as a candidate solution.The general solution process of the algorithm in GAO is explained in Algorithm 1.

Solution Process of the GAO
The proposed GAO approach is a biomimetics metaheuristic algorithm that mimics the natural behavior of the giant armadillo in the wild.Among the natural behaviors of the giant armadillo, the strategy of this animal in attacking termite mounds and then digging in them for feeding is employed in the GAO design.In this modeling, the wild life of the giant armadillo corresponds to the problem-solving space, and the position of each giant armadillo in the wild corresponds to the position of each GAO member in the problem-solving space as a candidate solution.The general solution process of the algorithm in GAO is explained in Algorithm 1.
Algorithm 1: Solution process of GAO Start.

1.
A certain number of giant armadillos are randomly initialized in the problem-solving space as a population of the algorithm, each representing a candidate solution for the problem.

2.
Based on the evaluation of each of the candidate solutions in the objective function and the comparison of the obtained values, the best GAO member is identified as the best candidate solution.

3.
In the first phase of the GAO, based on the modeling of the movement of the giant armadillo towards the termite mounds, the position of the GAO members in the problem-solving space and, as a result, the candidate solutions are updated.4.
In the second phase of GAO, based on the modeling of the small displacements of the giant armadillo while digging in termite mounds, the position of GAO members in the problem-solving space and, as a result, candidate solutions are updated.

5.
The third and fourth steps are repeated for all GAO members.6.
Based on the comparison of the new evaluated values for the objective function corresponding to the updated candidate solutions, the best candidate solution is identified, updated, and stored.

7.
The third to sixth steps are repeated until the last iteration of the algorithm.

8.
The best candidate solution obtained during the iterations of the algorithm is presented as the GAO solution for the given problem. End.
In the following, the solution process described for GAO is mathematically modeled in full.

Mathematical Modeling of GAO
In this subsection, the implementation steps of GAO are fully modeled.For this purpose, first, the initialization process of GAO has been explained and modeled.Then, the mathematical model of the process of updating candidate solutions in two phases of exploration and exploitation is presented.

Algorithm Initialization
The proposed GAO approach is a population-based meta-heuristic algorithm that assumes that giant armadillos form its population.GAO is able to provide suitable solutions for optimization problems in an iterative process based on the search power of its members in the problem-solving space.Each GAO member, based on his position in the problemsolving space, determines the values for the decision variables of the problem.Therefore, each giant armadillo, as a member of the population, is a candidate solution to the problem that is modeled from a mathematical point of view using a vector.Giant armadillos together form the population of the algorithm, which can be modeled from a mathematical point of view using a matrix according to Equation (1).The primary position of the giant armadillos in the problem-solving space is randomly initialized at the beginning of the algorithm execution using Equation (2).
Here, X is the GAO population matrix, X i is the ith GAO member (candidate solution), x i,d is its dth dimension in search space (decision variable), N is the number of giant armadillos, m is the number of decision variables, r is a random number in interval [0, 1], lb d , and ub d are the lower bound and upper bound of the dth.decision variable, respectively.
Since the position of each giant armadillo in the problem-solving space represents a candidate solution for the problem, a value for the objective function can be evaluated corresponding to each giant armadillo.According to this, the set of evaluated values for the objective function can be represented using Equation (3).
Here, F is the vector of the evaluated objective function, and F i is the evaluated objective function based on the ith GAO member.
The evaluated values for the objective function provide valuable information about the quality of the candidate solutions proposed by the population members.The best value obtained for the objective function corresponds to the best member (i.e., the best candidate solution), and the worst value obtained for the objective function corresponds to the worst member (i.e., the worst candidate solution).Since in each iteration, the position of the giant armadillos in the problem-solving space is updated, the best member should also be updated based on the comparison of the updated values for the objective function.At the end of the implementation of the algorithm, the position of the best member obtained during the iterations of the algorithm is presented as a solution to the problem.
In the design of the proposed GAO approach, the position of the population members in the problem-solving space is updated based on the modeling of the hunting strategy of giant armadillos in the wild.In this process, the giant armadillo first attacks the position of termite mounds, then digs in termite mounds to hunt and eat termites.According to this, in each iteration of GAO, the position of the population members is updated in two phases: (i) exploration, based on the simulation of the movement of giant armadillos towards termite mounds, and (ii) exploitation, based on the simulation of giant armadillos digging in termite mounds to feed on termites.

Phase 1: Attack on Termite Mounds (Exploration Phase)
In the first phase of GAO, the position of the population members in the problemsolving space is updated based on the simulation of the attack of the giant armadillo towards the termite mounds during hunting.In the GAO design, it is inspired by the changing position of the giant armadillo while moving towards the termite mounds in order to update the position of the population members in the problem-solving space.Modeling this attack process leads to extensive changes in the position of the giant armadillo and, as a result, increases the exploration power of the algorithm in global search management.
In the GAO design, for each population member that represents a giant armadillo, the location of other population members that have a better objective function value is considered a termite mound.The set of candidate termite mounds for each member of the population is specified using Equation (4).
Here, TM i is the set of candidate termite mounds' locations for the ith giant armadillo, X k is the population member with a better objective function value than the ith giant armadillo, and F k is its objective function value.
The giant armadillo randomly selects one of the candidate termite mounds and attacks it.Based on modeling the movement of giant armadilloes towards termite mounds, a new position is calculated for each member of the population using Equation (5).Then, this new position replaces the previous position of the corresponding member if it improves the value of the objective function according to Equation (6).
Here, STM i is the selected termite mound for ith giant armadillo, STM i,j is its jth dimension, X P1 i is the new position calculated for the ith giant armadillo based on attacking phase of the proposed GAO, x P1 i,j is its jth dimension, F P1 i is its objective function value, r i,j are random numbers from the interval [0, 1], and I i,j are numbers which are randomly selected as 1 or 2.

Phase 2: Digging in Termite Mounds (Exploitation Phase)
In the second phase of GAO, the position of population members in the problemsolving space is updated based on the simulation of giant armadillo digging in termite mounds to feed on termites.Modeling this giant armadillo digging process with the aim of hunting and eating termites leads to small changes in the position of the giant armadillo and, as a result, increases the exploitation power of the algorithm in local search management.
In the GAO design, based on modeling the skill of the giant armadillo to dig in termite mounds, a new position is calculated for each member of the population using Equation (7).Then, if the value of the objective function is improved, this new position replaces the previous position of the corresponding member according to Equation (8).
Here, X P2 i is the new position calculated for the ith giant armadillo based on digging phase of the proposed GAO, x P2 i,j is its jth dimension, F P2 i is its objective function value, r i,j are random numbers from the interval [0, 1], and t is the iteration counter.

Repetition Process, Pseudocode, and Flowchart of GAO
After updating the position of all giant armadillos in the problem-solving space based on the attack and digging phases, the first iteration of GAO is completed.After that, the algorithm enters the next iteration, and the process of updating the position of giant armadillos in the problem-solving space continues until the last iteration of the algorithm using Equations ( 4)- (8).In each iteration, the position of the best GAO member is updated and stored as the best candidate solution.After the full implementation of GAO on the given problem, the best candidate solution recorded during the iterations of the algorithm is presented as the solution to the problem.The implementation steps of GAO are presented as a flowchart in Figure 2, and its pseudocode is presented in Algorithm 2. The complete set of codes is available at the following repository: https://uk.mathworks.com/matlabcentral/fileexchange/156329-giant-armadillo-optimization (accessed on 13 November 2023).For t = 1 to T 6.
For i = 1 to N 7.
Determine the termite mounds set for the ith GAO member using Equation (4).
Select the termite mounds for the ith GAO member at random.10.
Calculate new position of ith GAO member using Equation (5).
Calculate new position of ith GAO member using Equation (7).
Save the best candidate solution so far.17. end 18.Output the best quasi-optimal solution obtained with the GAO.End GAO.

Computational Complexity of GAO
In this subsection, the computational complexity of the proposed GAO approach is evaluated.The preparation and initialization process of GAO has a computational complexity equal to O(Nm), where N is the number of giant armadillos and m is the number of decision variables of the problem.In the GAO design, in each iteration, the position of each giant armadillo is updated in two phases of exploration and exploitation.Therefore, the GAO update process has a computational complexity equal to O(2NmT), where T is the maximum number of iterations of the algorithm.According to this, the total computational complexity of the proposed GAO approach is equal to O(Nm(1 + 2T)).

Comparing GAO vs. PSO
In this subsection, the proposed GAO approach is compared with PSO.PSO is a well-known bio-inspired metaheuristic algorithm that has been used in many optimization applications by researchers.
In terms of the main design idea, PSO is inspired by the collective movement of groups of birds or fish that are searching for food.On the other hand, GAO was inspired by the giant armadillo's strategy of attacking termite mounds and digging to feed on them.So, the difference in the main design idea is evident.
In PSO, the position of each member of the population is updated according to the position of the best member of the population and the previous best position of the corresponding member.On the other hand, the position of each member of the population in the problem-solving space is updated based on the position of a better member (from the point of view of comparing the value of the objective function) and also based on local search management near each member's position.
A very important point in GAOs performance is that it has avoided a heavy dependence of the population update process on the best members.These conditions lead to the improvement of GAOs performance in global search management, preventing premature convergence, and preventing the algorithm from getting stuck in local optima.Meanwhile, in the design of PSO, the update process relies heavily on the position of the best member, which leads to inappropriate rapid convergence and stops the entire population from adopting a similar solution.
Another important point in the design of metaheuristic algorithms is the control parameters.Determining the values of control parameters is a challenging process, and for this reason, the design of parameter-less approaches is considered a major advantage.The mathematical model of PSO has three control parameters, the value of which has a significant impact on the performance of this algorithm.This is despite the fact that no control parameters are included in the design of GAO, and from this point of view, GAO is a parameter-less approach.

Simulation Studies and Results
In this section, GAOs performance in solving optimization problems is evaluated.For this purpose, the efficiency of GAO is tested in handling the CEC 2017 test suite for problem dimensions equal to 10, 30, 50, and 100.

Performance Comparison
In order to measure the effectiveness of GAO in solving optimization problems, the obtained results are compared with the performance of twelve famous metaheuristic algorithms: GA [35], PSO [18], GSA [42], TLBO [56], MVO [44], GWO [22], WOA [25], MPA [31], TSA [32], RSA [27], AVOA [86], and WSO [33].From the numerous optimization algorithms designed so far, these twelve methods have been selected for comparison with GAO.The reason for choosing these twelve competitor algorithms is that GA and PSO are the best-known and most widely used optimization algorithms.GSA, TLBO, MVO, and GWO, introduced between 2009 and 2016, have been popular methods for researchers and have been widely cited.WOA, MPA, and TSA algorithms are among the most widely used techniques introduced from 2016 to 2020.RSA, AVOA, and WSO are recently developed optimizers that have quickly gained the attention of scientists and have been used in a variety of real-world applications.The control parameter values of metaheuristic algorithms are specified in Appendix A and Table A1.The results of simulation studies are presented using six statistical indicators: mean, best, worst, standard deviation (std), median, and rank.The values obtained for the mean index are used as a ranking criterion for metaheuristic algorithms in handling each of the benchmark functions.

Evaluation of the CEC 2017 Test Suite
In this subsection, the performance of GAO and competitor algorithms is benchmarked in handling the CEC 2017 test suite for problem dimensions equal to 10, 30, 50, and 100.The CEC 2017 test suite has 30 standard benchmark functions consisting of (i) three unimodal functions of C17-F1 to C17-F3, (ii) seven multimodal functions of C17-F4 to C17-F10, (iii) ten hybrid functions of C17-F11 to C17-F20, and (iv) ten composition functions of C17-F21 to C17-F30.The C17-F2 functional is excluded from simulation studies due to its unstable behavior.Full information and more details about the CEC 2017 test suite are available at [87].
The implementation results of GAO and competitor algorithms on the CEC 2017 test suite are reported in Tables 1-4.Boxplot diagrams obtained from the performance of metaheuristic algorithms are drawn in Figures 3-6.Based on the analysis of the simulation results, the proposed GAO approach in handling the CEC 2017 test suite, for problem dimensions equal to 10 (m = 10), is the first best optimizer for functions C17-F1, C17-F3 to C17-F21, C17-F23, C17-F24, and C17-F27 to C17-F30 (i.e., 26 functions from 29 functions).Therefore, for problem dimensions equal to 10 (m = 10), GAO has been the first best optimizer in 26 out of 29 functions (i.e., 89.65% of test functions) and has provided superior performance compared to competing algorithms.
For problem dimensions equal to 50 (m = 50), the proposed GAO approach is the first best optimizer for functions C17-F1, C17-F3 to C17-F25, and C17-F27 to C17-F30.Therefore, for problem dimensions equal to 50 (m = 50), GAO has been the best optimizer in 28 out of 29 functions (i.e., 96.55% of test functions) and has provided superior performance compared to competing algorithms.
For problem dimensions equal to 100 (m = 100), the proposed GAO approach is the first best optimizer for functions C17-F1, C17-F3, and C17-F30.Therefore, for problem dimensions equal to 100 (m = 100), GAO has been the first best optimizer in 29 out of 29 functions (i.e., 100% of test functions) and has provided superior performance compared to competing algorithms.
The optimization results show that the proposed GAO approach has achieved good results for the benchmark functions, with high abilities in exploration, exploitation, and balance during the search process.What is clear from the simulation results is that GAO has provided superior performance by providing better results for most benchmark functions compared to competitor algorithms in dealing with the CEC 2017 test suite for problem dimensions equal to 10, 30, 50, and 100.

Statistical Analysis
In this subsection, using statistical analysis of the obtained results, it has been checked whether the superiority of the proposed GAO approach is significant from a statistical point of view or not.For this purpose, the Wilcoxon rank sum test [88] is employed, which is a non-parametric test and is used to determine the significant difference between the means of two data samples.In the Wilcoxon rank sum test, the presence or absence of a statistically significant difference is determined using an index called the p-value.The implementation results of the Wilcoxon rank sum test statistical analysis on the performance of GAO against each of the competitor algorithms are reported in Table 5.Based on the obtained results, in cases where the p-value is less than 0.05, GAO has a statistically significant superiority compared to the corresponding competitor algorithm.Statistical analysis shows that GAO has a significant statistical superiority in handling the CEC 2017 test suite for all four dimensions of the problem, equal to 10, 30, 50, and 100, in competition with all twelve compared algorithms.

GAO for Real-World Applications
In this section, the effectiveness of the proposed GAO approach in solving optimization problems in real-world applications is evaluated.For this purpose, twenty-two constrained optimization problems from the CEC 2011 test suite and four engineering design problems are selected.

Evaluation of the CEC 2011 Test Suite
In this subsection, the performance of GAO and competitor algorithms in handling the CEC 2011 test suite has been tested.The CEC 2011 test suite consists of twenty-two constrained optimization problems from real-world applications.A full description and more details about the CEC 2011 test suite are available at [89].
The results of employing GAO and competitor algorithms to deal with the CEC 2011 test suite are reported in Table 6.The boxplot diagrams obtained from the performance of metaheuristic algorithms in this experiment are plotted in Figure 7.The optimization results show that the proposed GAO approach, with its high ability to explore, exploit, and balance them during the search process, has been able to provide suitable solutions for optimization problems.What is concluded from the comparison of the simulation results is that GAO has provided superior performance in handling the CEC 2011 test suite against competitor algorithms by providing better results for most of the benchmark functions and obtaining the rank of the first-best optimizer overall.Also, the results obtained from the Wilcoxon rank sum test indicate the statistically significant superiority of GAO compared to all twelve competitor algorithms in order to solve the CEC 2011 test suite.

Pressure Vessel Design Problem
Pressure vessel design is a real-world engineering challenge with the aim of minimizing construction costs.The schematic of this design is shown in Figure 8, and its mathematical model is as follows [90]: Consider:  =  ,  ,  ,  =  ,  , ,  .Minimize: () = 0.6224   + 1.778  + 3.1661  + 19.84  .

Pressure Vessel Design Problem
Pressure vessel design is a real-world engineering challenge with the aim of minimizing construction costs.The schematic of this design is shown in Figure 8, and its mathematical model is as follows [90]   Consider: The implementation results of the GAO and competitor algorithms on the Pressure vessel design problem are reported in Tables 7 and 8.The convergence curve of GAO while achieving the optimal solution for pressure vessel design is drawn in Figure 9. Based on the optimization results, GAO has determined the optimal design for the pressure vessel with the values of the design variables equal to (0.7780271, 0.3845792, 40.312284, and 200) and the value of the objective function equal to (5882.8955).The simulation results show that GAO has provided superior performance in dealing with the pressure vessel design problem by providing better results compared to competitor algorithms.

Speed Reducer Design Problem
Speed reducer design is a real-world engineering challenge with the aim of minimizing the weight of the speed reducer.The schematic of this design is shown in Figure 10, and its mathematical model is as follows [91,92]

Speed Reducer Design Problem
Speed reducer design is a real-world engineering challenge with the aim of minimizing the weight of the speed reducer.The schematic of this design is shown in Figure 10, and its mathematical model is as follows [91,92]: indicates that GAO has provided superior performance by achieving better results in order to solve the problem of speed reducer design compared to competitor algorithms.Consider: X = x 1, x 2 , x 3 , x 4 , x 5 , x 6 , x 7 = b, m, p, l 1 , l 2 , d 1 , d 2 .
The results of employing GAO and competitor algorithms to solve the speed reducer design problem are reported in Tables 9 and 10.The convergence curve of GAO towards the optimal solution for speed reducer design is drawn in Figure 11.Based on the optimization results, GAO has provided the optimal design for the speed reducer with the values of the design variables equal to (3.5, 0.7, 17, 7.3, 7.8, 3.3502147, and 5.2866832) and the value of the objective function equal to (2996.3482).Analysis of the simulation results indicates that GAO has provided superior performance by achieving better results in order to solve the problem of speed reducer design compared to competitor algorithms.

Welded Beam Design
Welded beam design is a real-world engineering challenge with the aim of minimizing the fabrication cost of the welded beam.The schematic of this design is shown in Figure 12, and its mathematical model is as follows [25]: Consider:  =  ,  ,  ,  = ℎ, , ,  .Minimize: () = 1.10471  + 0.04811  (14.0 +  ).

Welded Beam Design
Welded beam design is a real-world engineering challenge with the aim of minimizing the fabrication cost of the welded beam.The schematic of this design is shown in Figure 12, and its mathematical model is as follows [25]:  Consider: X = [x 1 , x 2 , x 3 , x 4 ] = [h, l, t, b].Minimize: f (x) = 1.10471x 2 1 x 2 + 0.04811x 3 x 4 (14.0 + x 2 ).Subject to: g 1 (x) = τ(x) − 13, 600 ≤ 0, g 2 (x) = σ(x) − 30, 000 ≤ 0, g 3 (x) = x 1 − x 4 ≤ 0, g 4 (x) = 0.10471x 2  1 + 0.04811x 3 x 4 (14 + x 2 ) − 5.0 ≤ 0, g 5 (x) = 0.125 − x 1 ≤ 0, g 6 (x) = δ (x) − 0.25 ≤ 0,  The results of dealing with the problem of welded beam design using GAO and competitor algorithms are reported in Tables 11 and 12.The convergence curve of GAO while achieving the optimal solution for welded beam design is drawn in Figure 13.Based on the optimization results, GAO has determined the optimal design for the welded beam with the values of the design variables equal to (0.2057296, 3.4704887, 9.0366239, and 0.2057296) and the value of the objective function equal to (1.7246798).What is evident from the simulation results is that GAO has provided superior performance by converging to better results in order to address the welded beam design problem compared to competitor algorithms.

Tension/Compression Spring Design
Tension/compression spring design is a real-world engineering challenge with the aim of minimizing the weight of the tension/compression spring.The schematic of this design is shown in Figure 14, and its mathematical model is as follows [25]: Consider:  =  ,  ,  = , ,  .Minimize: () = ( + 2)  .The implementation results of GAO and competitor algorithms on the tension/compression spring design problem are reported in Tables 13 and 14.The convergence curve of GAO towards the optimal solution for tension/compression spring design is drawn in Figure 15.Based on the optimization results, GAO has determined the optimal design for the tension/compression spring with the values of the design variables equal to (0.0516891,

Tension/Compression Spring Design
Tension/compression spring design is a real-world engineering challenge with the aim of minimizing the weight of the tension/compression spring.The schematic of this design is shown in Figure 14, and its mathematical model is as follows [25]: 0.3567177, and 11.288966) and the value of the objective function equal to (0.0126019).The simulation results show that GAO has provided superior performance by providing better results for solving the tension/compression spring design problem compared to competitor algorithms.The implementation results of GAO and competitor algorithms on the tension/compression spring design problem are reported in Tables 13 and 14.The convergence curve of GAO towards the optimal solution for tension/compression spring design is drawn in Figure 15.Based on the optimization results, GAO has determined the optimal design for the tension/compression spring with the values of the design variables equal to (0.0516891, 0.3567177, and 11.288966) and the value of the objective function equal to (0.0126019).The simulation results show that GAO has provided superior performance by providing better results for solving the tension/compression spring design problem compared to competitor algorithms.

Conclusions and Future Works
In this paper, a new bio-inspired metaheuristic algorithm called Giant Armadillo Optimization (GAO) was introduced, which imitates the behavior of giant armadilloes in nature.The fundamental inspiration for GAOs design is derived from the attack strategy of giant armadillos in moving towards prey positions and digging termite mounds.The

Conclusions and Future Works
In this paper, a new bio-inspired metaheuristic algorithm called Giant Armadillo Optimization (GAO) was introduced, which imitates the behavior of giant armadilloes in nature.The fundamental inspiration for GAOs design is derived from the attack strategy of giant armadillos in moving towards prey positions and digging termite mounds.The GAO theory was stated, and its implementation steps were mathematically modeled in two phases: (i) exploration based on the simulation of the movement of giant armadillos towards termite mounds, and (ii) exploitation based on the simulation of the giant armadillo's digging skills in order to prey on and rip open termite mounds.The efficiency of GAO was evaluated in handling the CEC 2017 test suite for problem dimensions equal to 10, 30, 50, and 100.The optimization results showed that GAO has a high ability for exploration, exploitation, and balancing them during the search process.The results obtained from GAO were compared with the performance of twelve well-known metaheuristic algorithms.The simulation results showed that GAO has provided superior performance by achieving better results for most of the benchmark functions in competition with competitor algorithms.Using the statistical analysis of the Wilcoxon rank sum test, it was confirmed that GAO has a significant statistical superiority over competitor algorithms.Implementation of GAO on the CEC 2011 test suite and four engineering design problems showed that the proposed approach has an effective ability to handle optimization tasks in real-world applications.
Introducing the proposed GAO approach raises several research tasks for further work.
• Binary GAO.The real version of GAO is introduced and fully designed in this paper.However, many optimization problems in science, such as feature selection, should be optimized using binary versions of metaheuristic algorithms.According to this, designing the binary version of the proposed GAO approach (BGAO) is one of the special potentials of this study.

•
Multi-objective GAO.From the point of view of the number of objective functions, optimization problems are divided into single-objective and multi-objective categories.
In many optimization problems, several objective functions must be considered simultaneously in order to achieve a suitable solution.Therefore, developing the multi-objective version of the proposed GAO approach (MOGAO) in order to handle multi-objective optimization problems is another research potential of this paper.

•
Hybrid GAO.Combining two or more metaheuristic algorithms in order to benefit from the advantages of each algorithm and create an effective hybrid approach has always been of interest to researchers.Considering this, developing hybrid versions of the proposed GAO approach is another research proposal for future work.

•
Tackle new domains.GAO employment to address real-world applications and optimization problems in various sciences such as renewable energy, chemical engineering, robotics, and image processing are among other research proposals for further work.

Figure 1 .
Figure 1.Giant armadillo taken from: free media Wikimedia Commons.

Figure 1 .
Figure 1.Giant armadillo taken from: free media Wikimedia Commons.

Figure 7 .
Figure 7. Boxplot diagrams of GAO and competitor algorithmsʹ performances on the CEC 2011 test suite.

Figure 7 .
Figure 7. Boxplot diagrams of GAO and competitor algorithms' performances on the CEC 2011 test suite.

Figure 8 .
Figure 8. Schematic of pressure vessel design.Figure 8. Schematic of pressure vessel design.

Figure 8 .
Figure 8. Schematic of pressure vessel design.Figure 8. Schematic of pressure vessel design.

Figure 9 .
Figure 9. GAOs performance convergence curve on pressure vessel design.

Figure 9 .
Figure 9. GAOs performance convergence curve on pressure vessel design.

Figure 10 .
Figure 10.Schematic of speed reducer design.Figure 10.Schematic of speed reducer design.

Figure 10 .
Figure 10.Schematic of speed reducer design.Figure 10.Schematic of speed reducer design.

Figure 12 .
Figure 12.Schematic of welded beam design.Figure 12. Schematic of welded beam design.

Figure 14 .
Figure 14.Schematic of tension/compression spring design.Figure 14.Schematic of tension/compression spring design.

Table 5 .
Wilcoxon rank sum test results.

Table 6 .
Optimization results of the CEC 2011 test suite.

Table 7 .
Performance of optimization algorithms on pressure vessel design problem.

Table 8 .
Statistical results of optimization algorithms on pressure vessel design problem.

Table 9 .
Performance of optimization algorithms on speed reducer design problem.

Table 10 .
Statistical results of optimization algorithms on speed reducer design problem.

Table 9 .
Performance of optimization algorithms on speed reducer design problem.

Table 10 .
Statistical results of optimization algorithms on speed reducer design problem.

Table 11 .
Performance of optimization algorithms on welded beam design problem.

Table 12 .
Statistical results of optimization algorithms on welded beam design problem.

Table 13 .
Performance of optimization algorithms on tension/compression spring design problem.

Table 14 .
Statistical results of optimization algorithms on tension/compression spring design problem.

Table 13 .
Performance of optimization algorithms on tension/compression spring design problem.

Table 14 .
Statistical results of optimization algorithms on tension/compression spring design problem.